Gibbs and Helmholtz energies

The second law decides the direction of change by the total entropy of a system and its surroundings. That criterion is correct but inconvenient, because it forces us to think about the surroundings every time. Chemists almost always work at constant temperature, and usually at constant pressure as well, and under those conditions the entropy change of the surroundings can be expressed entirely in terms of the system. The result is two new state functions, the Helmholtz energy and the Gibbs energy, which decide spontaneity from properties of the system alone and also tell us the maximum work that a process can deliver. The Gibbs energy is the single most important quantity in chemical thermodynamics, and the rest of this course is built on it.

From total entropy to the system alone

Consider a system at constant temperature T and constant pressure, in contact with surroundings at the same temperature. Lesson 5 showed that \(\Delta S_\mathrm{surr} = -\Delta H/T\), where ΔH is the enthalpy change of the system. The second law requires \(\Delta S_\mathrm{sys} + \Delta S_\mathrm{surr} \geq 0\) for any allowed change, so \(\Delta S - \Delta H/T \geq 0\), where the unlabeled quantities refer to the system. Multiplying by −T reverses the inequality: \(\Delta H - T\Delta S \leq 0\). This suggests defining the Gibbs energy

\[G = H - TS, \qquad \Delta G = \Delta H - T\Delta S \ (\text{constant } T), \qquad \Delta G = -T\Delta S_\mathrm{tot}.\]

At constant temperature and pressure, a process is spontaneous if \(\Delta G < 0\), at equilibrium if \(\Delta G = 0\), and not spontaneous in the forward direction if \(\Delta G > 0\); in that case the reverse process is spontaneous. A decrease in the Gibbs energy of the system is exactly an increase in the total entropy, scaled by −T. The same argument at constant temperature and volume, where the heat exchanged is ΔU, gives the Helmholtz energy \(A = U - TS\), which decreases in any spontaneous change at constant T and V.

Josiah Willard Gibbs introduced the function now named after him in his paper On the Equilibrium of Heterogeneous Substances, published in the Transactions of the Connecticut Academy in two parts in 1876 and 1878. Gibbs worked alone at Yale, wrote in a dense mathematical style and published in an obscure journal, so his work took years to reach chemists in Europe; Wilhelm Ostwald translated it into German in 1892 and Henry Le Chatelier into French in 1899. Hermann von Helmholtz introduced the free energy A independently in 1882, in a paper on the thermodynamics of chemical processes in galvanic cells. The older names free energy and Gibbs free energy are still common; IUPAC recommends Gibbs energy and Helmholtz energy, and the symbol F is sometimes used for A, especially in physics.

Maximum work

The Gibbs and Helmholtz energies do more than decide direction. They measure the work a process can do. For a reversible change at constant temperature, \(\delta q_\mathrm{rev} = T\,\mathrm{d}S\), and the first law gives \(\mathrm{d}U = T\,\mathrm{d}S + \delta w_\mathrm{rev}\), so \(\mathrm{d}A = \mathrm{d}U - T\,\mathrm{d}S = \delta w_\mathrm{rev}\). The work done on the system in a reversible isothermal change equals ΔA, and the maximum work that the system can do on its surroundings is \(-\Delta A\). That is the origin of the German name Arbeit, work, and of the symbol A.

At constant pressure, part of the work is unavoidably spent pushing back the atmosphere. The work beyond expansion work, called non-expansion or additional work, is what usually matters: the electrical work of a battery, the mechanical work of a muscle, the chemical work of building a protein. Repeating the argument with \(\delta w = -P\,\mathrm{d}V + \delta w_\mathrm{add}\) shows that at constant temperature and pressure \(\mathrm{d}G = \delta w_\mathrm{add,rev}\). The maximum non-expansion work a process can deliver is \(-\Delta G\). A hydrogen–oxygen fuel cell producing liquid water has \(\Delta_\mathrm{r}G^{\circ} = -237.1\ \mathrm{kJ\,mol^{-1}}\) and \(\Delta_\mathrm{r}H^{\circ} = -285.8\ \mathrm{kJ\,mol^{-1}}\), so at most 237.1 kJ of each 285.8 kJ can appear as electrical work, about 83 parts in 100. The rest must be released as heat, \(T\Delta S\), because the reaction lowers the entropy of the system.

Standard Gibbs energies

The standard Gibbs energy of reaction \(\Delta_\mathrm{r}G^{\circ}\) can be found in two ways: from \(\Delta_\mathrm{r}H^{\circ} - T\Delta_\mathrm{r}S^{\circ}\), using enthalpies of formation and absolute entropies, or from tabulated standard Gibbs energies of formation, \(\Delta_\mathrm{f}G^{\circ}\), the Gibbs energy change for forming one mole of a compound from its elements in their reference states. As with enthalpies, \(\Delta_\mathrm{f}G^{\circ} = 0\) for every element in its reference state, and \(\Delta_\mathrm{r}G^{\circ} = \sum\nu\Delta_\mathrm{f}G^{\circ}(\text{products}) - \sum\nu\Delta_\mathrm{f}G^{\circ}(\text{reactants})\).

A compound with a negative Gibbs energy of formation is thermodynamically stable with respect to its elements; one with a positive value, such as nitrogen dioxide, could in principle decompose into its elements spontaneously, and persists only because the decomposition is slow. Diamond has ΔfG° = +2.90 kJ/mol, so at room temperature and 1 bar it is less stable than graphite; it survives because its conversion to graphite has an enormous activation barrier.

Enthalpy, entropy and temperature

Because \(\Delta G = \Delta H - T\Delta S\), the sign of ΔG depends on the signs of ΔH and ΔS and on the temperature. If ΔH and ΔS change little with temperature, which is often a fair approximation over a few hundred kelvin, the four combinations behave as lesson 6 described. When ΔH and ΔS have the same sign, there is a crossover temperature \(T^{*} = \Delta H/\Delta S\) at which \(\Delta G = 0\) and the reaction changes from spontaneous to non-spontaneous under standard conditions or the reverse. For the decomposition of limestone, \(T^{*} = 178\,300/159.0 = 1121\ \mathrm{K}\), about 850 °C, which is close to the temperature at which lime kilns operate; above it, the pressure of carbon dioxide over the solids exceeds 1 bar. For ammonia synthesis, \(T^{*} \approx 463\ \mathrm{K}\).

How G depends on temperature and pressure

Differentiating \(G = H - TS\) and using \(\mathrm{d}H = T\,\mathrm{d}S + V\,\mathrm{d}P\), which lesson 8 derives, gives \(\mathrm{d}G = V\,\mathrm{d}P - S\,\mathrm{d}T\). Two consequences follow at once. First, \((\partial G/\partial T)_P = -S\): because entropy is always positive, the Gibbs energy of every substance falls as the temperature rises, and it falls fastest for gases, which have the largest entropies. That is why heating favors the phase or the side of a reaction with more entropy. A closely related result, the Gibbs–Helmholtz equation,

\[\Bigl(\frac{\partial (\Delta G/T)}{\partial T}\Bigr)_P = -\frac{\Delta H}{T^2},\]

shows that the temperature dependence of \(\Delta G/T\) is governed by ΔH. Lesson 11 uses it to derive the van 't Hoff equation for the temperature dependence of equilibrium constants. Second, \((\partial G/\partial P)_T = V\): raising the pressure raises the Gibbs energy of every substance, and most for gases, which have the largest volumes. For a liquid or solid, whose volume hardly changes with pressure, \(G(P_2) \approx G(P_1) + V\Delta P\). For an ideal gas, integrating \(V = nRT/P\) gives

\[G_\mathrm{m}(P) = G_\mathrm{m}^{\circ} + RT\ln\frac{P}{P^{\circ}}.\]

This logarithmic dependence on pressure is the seed of the whole theory of equilibrium. Lesson 8 generalizes it to the chemical potential of a component in a mixture, and lesson 11 turns it into the equilibrium constant.

Diamonds from graphite

Pressure favors the denser phase, because the denser phase has the smaller molar volume and so its Gibbs energy rises more slowly with pressure. Diamond, with a density of 3.51 g/cm³, is much denser than graphite at 2.27 g/cm³. Their molar volumes are 3.42 and 5.30 cm³/mol, a difference of −1.88 cm³/mol. The Gibbs energy of conversion of graphite to diamond, +2.90 kJ/mol at 1 bar, therefore falls by 1.88 J for every megapascal of extra pressure and reaches zero near 1.5 GPa, about 15 000 bar, at 298 K. At higher temperatures the required pressure is larger, because graphite has more entropy than diamond. In 1954 a team at General Electric made the first reproducible synthetic diamonds at pressures of several gigapascals and temperatures above 1500 °C, with a molten metal catalyst to overcome the kinetic barrier. The prediction of the required conditions from thermodynamic data was one of the most striking practical successes of the Gibbs energy.

Coupled reactions in living cells

A reaction with a positive ΔG cannot proceed on its own, but it can be driven by coupling it to one with a more negative ΔG, so that the combined process has ΔG < 0. Living cells do this constantly. The hydrolysis of ATP to ADP and phosphate has a standard Gibbs energy of about −30.5 kJ/mol under biochemical standard conditions, and considerably more negative values, around −50 kJ/mol, under the concentrations found in cells. Enzymes couple this hydrolysis to unfavorable steps such as the formation of peptide bonds or the pumping of ions across membranes. The Gibbs energy released by the oxidation of glucose, −2870 kJ/mol under standard conditions, is used to regenerate ATP from ADP. The bookkeeping of metabolism is Gibbs-energy bookkeeping.

A worked example: when does limestone decompose?

Lime production is one of the largest industrial sources of carbon dioxide, because every mole of lime releases a mole of CO₂ from the carbonate as well as the CO₂ from the fuel that heats the kiln. The same reaction runs in cement kilns, and the thermodynamics explains why it cannot be made to run at much lower temperatures: the entropy of the released gas is what drives it, and that drive becomes large enough only when T is large.

A program: Gibbs energy and the spontaneity temperature

The program computes ΔrG° from ΔrH° and ΔrS° over a range of temperatures for several reactions, finds each crossover temperature, and compares ΔrG°(298 K) with the value from Gibbs energies of formation. It then finds the pressure at which diamond becomes more stable than graphite at 298 K and at higher temperatures, using the molar volumes and an approximate entropy difference.

rxns = {   # dH (kJ), dS (J/K), dG from dfG tables (kJ), all at 298.15 K
    'N2 + 3 H2 -> 2 NH3':          (-91.8, -198.1, -32.8),
    'CaCO3 -> CaO + CO2':          (178.3, 159.0, 130.4),
    'N2O4 -> 2 NO2':               (57.2, 175.9, 4.7),
    'H2O(s) -> H2O(l)':            (6.01, 22.0, None),
    'H2 + 1/2 O2 -> H2O(l)':       (-285.8, -163.4, -237.1),
}
print(f'{"reaction":28s}{"dG(298)":>9s}{"tables":>8s}{"T* (K)":>9s}   dG at 500, 1000, 1500 K')
for name, (dH, dS, dGt) in rxns.items():
    dG = lambda T: dH - T * dS / 1000
    Tstar = dH * 1000 / dS if dH * dS > 0 else None
    ts = f'{Tstar:9.0f}' if Tstar else '     none'
    tab = f'{dGt:8.1f}' if dGt is not None else '       -'
    print(f'{name:28s}{dG(298.15):9.1f}{tab}{ts}   ' + '  '.join(f'{dG(T):7.1f}' for T in (500, 1000, 1500)))

# graphite -> diamond
dG0 = 2900.0                          # J/mol at 298 K and 1 bar
dV = (12.011 / 3.513 - 12.011 / 2.266) * 1e-6   # m^3/mol
dS = 2.38 - 5.74                     # J/(K mol), diamond minus graphite
print()
print(f'molar volume change graphite -> diamond: {dV * 1e6:.3f} cm3/mol')
for T in (298.15, 1000, 1500):
    dGT = dG0 - (T - 298.15) * dS    # (dG/dT)_P = -dS, with dS assumed constant
    P = 1e5 + dGT / (-dV)            # pressure at which dG = 0
    print(f'T = {T:7.1f} K  dG(1 bar) = {dGT / 1000:5.2f} kJ/mol  equilibrium pressure = {P / 1e9:4.2f} GPa')

reaction                      dG(298)  tables   T* (K)   dG at 500, 1000, 1500 K
N2 + 3 H2 -> 2 NH3              -32.7   -32.8      463       7.2    106.3    205.3
CaCO3 -> CaO + CO2              130.9   130.4     1121      98.8     19.3    -60.2
N2O4 -> 2 NO2                     4.8     4.7      325     -30.8   -118.7   -206.7
H2O(s) -> H2O(l)                 -0.5       -      273      -5.0    -16.0    -27.0
H2 + 1/2 O2 -> H2O(l)          -237.1  -237.1     1749    -204.1   -122.4    -40.7

molar volume change graphite -> diamond: -1.882 cm3/mol
T =   298.1 K  dG(1 bar) =  2.90 kJ/mol  equilibrium pressure = 1.54 GPa
T =  1000.0 K  dG(1 bar) =  5.26 kJ/mol  equilibrium pressure = 2.79 GPa
T =  1500.0 K  dG(1 bar) =  6.94 kJ/mol  equilibrium pressure = 3.69 GPa

The crossover temperatures are 463 K for ammonia synthesis, 1121 K for limestone, 325 K for the dissociation of dinitrogen tetroxide and 273 K for the melting of ice, the last one a good check that ΔfusH and ΔfusS are consistent. Hydrogen combustion, exothermic with an entropy decrease, also has a crossover, but only at about 1750 K; below it ΔrG° is negative, although it becomes less negative as the temperature rises. The diamond calculation gives an equilibrium pressure of about 1.5 GPa at room temperature, rising with temperature. The measured graphite–diamond boundary lies somewhat higher, near 2 GPa at room temperature and 5 GPa at 1500 K, because these estimates neglect the compressibility of the solids and the change of ΔS with temperature.